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Review Videos • • • • Graphing the x and y intercept Graphing the x and y intercepts Graphing a line in slope intercept form Converting into slope intercept form Chapter 7 Section 6 Families of Linear Graphs What You’ll Learn You’ll learn to explore the effects of changing the slopes and y-intercepts of linear functions. Why It’s Important Business Families of graphs can display different fees. Families of linear graphs often fall into two categories1. Those with the same slope 2. Those with the same y-intercept. Family of Graphs What do these lines have in common? y = ½x + 3 y = ½x - 1 Same Slope Family of Graphs What do these lines have in common? y = -x + 1 Same y-intercept y = ⅓x + 1 Not a Family of Graphs What do these lines have in common? y=x+2 y = ⅓x Different y-intercept and slope Example 1 Graph each pair of equations. Describe any similarities or differences. Explain why they are a family of graphs. y = 3x + 4 y = 3x – 2 The graphs have y-intercepts y = 3x + 4 of 4 and -2, respectively. They are a family of graphs because the slope of each line is 3. y = 3x - 2 Example 2 Graph each pair of equations. Describe any similarities or differences. Explain why they are a family of graphs. y=x+3 y = -½x + 3 Each graph has a different slope. y = -½x + 3 Each graph has a y-intercept of 3. Thus, they are a family of graphs. y=x+3 Hint: You can compare graphs of lines by looking at their equations. Example 3 • Matthew and Juan are starting their own pet care business. Juan wants to charge $5 an hour. Matthew thinks they should charge $3 an hour. Suppose x represents the number of hours. Then y = 5x and y = 3x represents how much they would charge, respectively. Compare and contrast the graphs of the equations. 6 The equations have the same yintercept, but the graph of y = 5x is steeper. This is because its slope, which represents $5 per hour, is greater that the slope of the graph of y = 3x. 5 y = 5x y = 3x 4 3 2 0 .5 1 1.5 2 Your Turn Compare and contrast the graphs of the equations. Verify by graphing the equation. y = -3x + 4 y = -x + 4 Answer y = -3x + 4 y = -x + 4 Same y-intercept Different slope y = -3x + 4 y = -x + 4 Try This One Compare and contrast the graphs of the equations. Verify by graphing the equation. y = ⅔x + 3 y = ⅔x -1 Answer y = ⅔x + 3 y = ⅔x -1 y = ⅔x + 3 Same slope Different y-intercept y = ⅔x - 1 Parent Graph A parent graph is the simplest of the graphs in a family. Let’s summarize how changing the m or b in y = mx + b affects the graph of the equation. Parent: y = x As the value of m Increases, the line Gets steeper y = 3x y=x y = ¼x Parent: y = -x y = -3x y = -x As the value of m Decreases, the line Gets steeper. y = -¼x Parent: y = 2x y = 2x y = 2x + 3 y = 2x - 4 As the value of b Increases, the graph shifts Up on the y-axis. As the value of b decreases, the graph shifts down on the y-axis You can change a graph by changing the slope or y-intercept. Example 4 Change y = -½x + 3 so that the graph of the new equation fits each description. y = -2x + 3 Same y-intercept, steeper negative slope The y-intercept is 3, and the slope is -½. The new equation will also have a y-intercept of 3. In order for the slope to be steeper and still be negative, its value must be less than -½, such as -2. The new equation is y = -2x + 3. y = -½x + 3 Example 5 Change y = -½x + 3 so that the graph of the new equation fits each description. y = -½x + 7 Same slope, y-intercept is shifted up 4 units The slope of the new equation will be -½. Since the current y-intercept will be 3 + 4 or 7. The new equation is y = -½x + 7. Always check by graphing y = -½x + 3 Your Turn Change y = 2x + 1 so that the graph of the new equation fits each description. Same slope, shifted down 1 unit. y = 2x + 0 Simplified to y = 2x Your Turn Change y = 2x + 1 so that the graph of the new equation fits each description. Same y-intercept, less steep positive slope y=x+1